I would prefer c-bet since I think players do not mess around too much on Kxx boards and hands that they might c/r flop are likely only flushdraws and sometimes KQ and KK/AA if he slowplayed pf.
So I think I at last b/c flop and decide on the turn if I want to call him down
If you c-bet flop then I would check back turn and call river bet, since I don´t want to b/c turn with my equity and he can turn smt worse into bluff or check it down his middle pp
As played: you have to call river since you have shown so much weakness that he might bet river with any pair or try to bluff smt like AT/AQ